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I am doing some tests with strictly increasing integer sequences whose gaps between consecutive elements show a "pseudorandom" behavior, meaning "pseudorandom" that the gaps do not grow up continuously, but they change from a bigger value to a smaller one and vice versa due to the properties of the sequence without an easy way of calculating those variations. Initially I am using the following sequences as an example:

  1. Prime numbers.

  2. Abundant numbers.

  3. Even deficient numbers.

  4. The natural numbers associated to the separated Möbius sequences $M_1$={Möbius $\mu(n)=-1$}, $M_2$={Möbius $\mu(n)=1$}, $M_3$={Möbius $\mu(n)=0$}

To continue with my tests I would require some other good examples, but I can not recall any other well known sequences with that behavior (not related with the ones above, or combinations of them).

Are there any other well known sequences in which the behavior of the consecutive gaps is "pseudorandom" in the way expressed here? Thank you!

(*) The reason of this question is the test explained here.

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iadvd
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    Sum of prime factors. – barak manos Apr 05 '16 at 05:04
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    @barakmanos thank you and my apologies, while you wrote the comment I added to clarify that should be strictly increasing, as all the samples I wrote. – iadvd Apr 05 '16 at 05:09
  • How exactly are the gaps between prime numbers "strictly increasing"??? – barak manos Apr 05 '16 at 05:10
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    @barakmanos strictly increasing integer sequences whose gaps between the consecutive elements are pseudorandom... the "strictly increasing integer sequence" is the prime number sequence itself... 2,3,5,7, etc. as the rest of sequences of the list. – iadvd Apr 05 '16 at 05:14
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    In that case, perfect numbers. – barak manos Apr 05 '16 at 05:18
  • @barakmanos thanks! that one was in my head too, but I was looking for other sequences not related with the ones above or combinations of them as I wrote in the last paragraph. The set of perfect numbers is the set of natural numbers removing the sets of abundant and deficient numbers. I was looking for other sequences not directly related with the ones listed above. – iadvd Apr 05 '16 at 05:25

3 Answers3

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Consider the sequence $a_n=\lfloor{r\cdot{b^n}}\rfloor$, where:

  • $r$ is any irrational number $>1$
  • $b$ is any natural number $>1$

For example, for $a_n=\lfloor\pi\cdot10^n\rfloor$ we get:

  • $a_0=3$
  • $a_1=31$
  • $a_2=314$
  • $a_3=3141$
  • $a_4=31415$
  • $a_5=314159$
  • $a_6=3141592$
  • $a_7=31415926$
  • $a_8=314159265$
  • $a_9=3141592653$
  • $\dots$

For generally smaller gaps, use a generally small value of $b$.

barak manos
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    I think OP would like things where the gap size does not blow up so much. Maybe take your idea mod some number of appropriate size, and use the fact that the multiples of pi is dense in the circle or something. – A. Thomas Yerger Apr 06 '16 at 06:40
  • @AlfredYerger: Gaps between prime numbers also eventually "blow up" (well, you should really define "blow up" mathematically, but in general, there is no limit on the gap between two consecutive prime numbers). BTW, my alternative suggestion at the bottom of the answer is not correct. For example, consider the irrational number $0.102001000200001000002000000\dots$. So I will remove it shortly. – barak manos Apr 06 '16 at 06:44
  • I don't think I was talking about prime number gaps. I just meant take your sequence mod some number of whatever size OP wants the values of the sequence to be in. Then maybe you can show that since the values of the multiples of pi mod 1 are dense in the interval, you can show that the values of $10^k \pi$ mod $c$ are dense in $[0,c]$. Then this should also satisfy OP's request. – A. Thomas Yerger Apr 06 '16 at 06:46
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    @AlfredYerger: You weren't, but OP has mentioned it as an example of what he/she are looking for. – barak manos Apr 06 '16 at 06:47
  • My suggestion is inappropriate anyway. The OP wants strictly increasing sequences. – A. Thomas Yerger Apr 06 '16 at 06:49
  • @AlfredYerger: Yes, I initially suggested "Sum of prime factors", but OP has ruled it out due to that reason. See the comment-thread on the question itself. – barak manos Apr 06 '16 at 06:51
  • @barakmanos thank you for the suggestion, indeed I would need not so bigger gaps as Alfred commented, but as you mention it is possible to use a generally small value of b. The only point I miss is that I am asking about "any other well known sequences" meaning that they are not invented but known in some field of Mathematics. – iadvd Apr 06 '16 at 09:38
  • @iadvd: The "randomness" of the decimal representation (or any other natural-base representation for that matter) of irrational numbers is very much known in some field of Mathematics (not sure what that field is called though, probably "algebra" or something similar). – barak manos Apr 06 '16 at 10:38
  • @barakmanos again thanks for your feedback, I was able to finish some tests, the reason I am looking for this kind of sequences is explained in this new question regarding Hamiltonian paths and finite graphs attached to the gaps of this kind of pseudorandom sequences. Just in case: http://math.stackexchange.com/questions/1731658/is-there-a-relationship-between-local-prime-gaps-and-cyclical-graphs – iadvd Apr 07 '16 at 07:31
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How about numbers that are the product of two distinct primes. 6, 10, 14, 15, 21, 22, ...

paw88789
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    From OP's comment on the question itself (#6 in the comment-thread): "looking for other sequences not related with the ones above or combinations of them as I wrote in the last paragraph". – barak manos Apr 06 '16 at 06:49
  • @paw88789 thank you! as Barak said I am looking for another sequences not related with the ones I listed as stated in the question. – iadvd Apr 06 '16 at 09:35
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Finally I found another way of obtaining this kind of sequences! Reviewing at OEIS, some kind of partition problems provide sequences whose gaps show also this kind of pseudorandom behavior I was looking for. For instance:

  1. The strictly increasing elements of the multiplicative partition function: number of ways of factoring n with all factors greater than 1. Taking only the elements strictly increasing it looks like: {1,2,3,4,5,7,9,12,16,19...} and the gaps are not strictly increasing, sometimes are bigger or lower depending on the properties of the sequence.

  2. Number of partitions of n into parts 5k+1 or 5k+4. Taking only the elements strictly increasing it looks like: {1,2,3,4,5,7,9,10,12,14,17,19,23...} and the gaps show the same pseudorandom behavior.

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